Nevertheless, theories with a non–singular bounce usually require either a violation of the null–energy condition or non–zero curvature [1]}, [2]}. A number of papers have studied bouncing cosmologies in the context of scalar–tensor theories, see e.g. [3]}, [4]}, [5]}. The model we will discuss in this paper relies on the assumption that our universe is spatially closed and takes its inspiration from [6]}Recent analyses of Planck data are consistent with a small positive curvature [7]}, [8]}, [9]}. In our model the spatial curvature at the present can indeed be very close to zero.. It relies on a modified gravity theory with a non–minimally coupled scalar field. The scalar field is assumed to sit in the minimum of an effective potential initially. Inflation after the bounce is driven by corrections to the Einstein–Hilbert action, which we assume to be of the form of \(R + R^2\)  gravityBouncing cosmologies in these type of modified gravity theories have been discussed in [10]}, [11]}..
We find that for a large range of parameter, the scalar field rolls down towards the minimum of an effective potential, thereby triggering a bounce, while the scalaron is driven up its potential energy, resulting in a period of Starobinsky–inflation. Our model is very similar to [6]}, with the addition of the dynamics of the scalaron field. The main result of our paper is that inflation driven by the \(R^2\) –terms happens naturally in such a setup and we provide evidence that the model is stable for a large region of parameter space.
